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///////////////////////////////////////////////////////////////////////////////
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// Copyright 2014 Anton Bikineev
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// Copyright 2014 Christopher Kormanyos
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// Copyright 2014 John Maddock
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// Copyright 2014 Paul Bristow
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// Distributed under the Boost
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// Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_HYPERGEOMETRIC_1F1_RECURRENCE_HPP_
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#define BOOST_HYPERGEOMETRIC_1F1_RECURRENCE_HPP_
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#include <boost/math/special_functions/modf.hpp>
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#include <boost/math/special_functions/next.hpp>
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#include <boost/math/tools/recurrence.hpp>
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#include <boost/math/special_functions/detail/hypergeometric_pFq_checked_series.hpp>
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namespace boost { namespace math { namespace detail {
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// forward declaration for initial values
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template <class T, class Policy>
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inline T hypergeometric_1F1_imp(const T& a, const T& b, const T& z, const Policy& pol);
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template <class T, class Policy>
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inline T hypergeometric_1F1_imp(const T& a, const T& b, const T& z, const Policy& pol, int& log_scaling);
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template <class T>
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struct hypergeometric_1F1_recurrence_a_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_a_coefficients(const T& a, const T& b, const T& z):
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a(a), b(b), z(z)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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const T ai = a + i;
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const T an = b - ai;
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const T bn = (2 * ai - b + z);
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const T cn = -ai;
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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hypergeometric_1F1_recurrence_a_coefficients operator=(const hypergeometric_1F1_recurrence_a_coefficients&);
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};
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template <class T>
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struct hypergeometric_1F1_recurrence_b_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_b_coefficients(const T& a, const T& b, const T& z):
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a(a), b(b), z(z)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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const T bi = b + i;
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const T an = bi * (bi - 1);
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const T bn = bi * (1 - bi - z);
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const T cn = z * (bi - a);
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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hypergeometric_1F1_recurrence_b_coefficients& operator=(const hypergeometric_1F1_recurrence_b_coefficients&);
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};
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//
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// for use when we're recursing to a small b:
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//
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template <class T>
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struct hypergeometric_1F1_recurrence_small_b_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_small_b_coefficients(const T& a, const T& b, const T& z, int N) :
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a(a), b(b), z(z), N(N)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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const T bi = b + (i + N);
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const T bi_minus_1 = b + (i + N - 1);
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const T an = bi * bi_minus_1;
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const T bn = bi * (-bi_minus_1 - z);
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const T cn = z * (bi - a);
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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hypergeometric_1F1_recurrence_small_b_coefficients operator=(const hypergeometric_1F1_recurrence_small_b_coefficients&);
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const T a, b, z;
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int N;
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};
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template <class T>
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struct hypergeometric_1F1_recurrence_a_and_b_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_a_and_b_coefficients(const T& a, const T& b, const T& z, int offset = 0):
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a(a), b(b), z(z), offset(offset)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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const T ai = a + (offset + i);
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const T bi = b + (offset + i);
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const T an = bi * (b + (offset + i - 1));
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const T bn = bi * (z - (b + (offset + i - 1)));
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const T cn = -ai * z;
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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int offset;
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hypergeometric_1F1_recurrence_a_and_b_coefficients operator=(const hypergeometric_1F1_recurrence_a_and_b_coefficients&);
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};
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#if 0
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//
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// These next few recurrence relations are archived for future reference, some of them are novel, though all
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// are trivially derived from the existing well known relations:
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//
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// Recurrence relation for double-stepping on both a and b:
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// - b(b-1)(b-2) / (2-b+z) M(a-2,b-2,z) + [b(a-1)z / (2-b+z) + b(1-b+z) + abz(b+1) /(b+1)(z-b)] M(a,b,z) - a(a+1)z^2 / (b+1)(z-b) M(a+2,b+2,z)
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//
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template <class T>
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struct hypergeometric_1F1_recurrence_2a_and_2b_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_2a_and_2b_coefficients(const T& a, const T& b, const T& z, int offset = 0) :
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a(a), b(b), z(z), offset(offset)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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i *= 2;
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const T ai = a + (offset + i);
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const T bi = b + (offset + i);
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const T an = -bi * (b + (offset + i - 1)) * (b + (offset + i - 2)) / (-(b + (offset + i - 2)) + z);
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const T bn = bi * (a + (offset + i - 1)) * z / (z - (b + (offset + i - 2)))
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+ bi * (z - (b + (offset + i - 1)))
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+ ai * bi * z * (b + (offset + i + 1)) / ((b + (offset + i + 1)) * (z - bi));
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const T cn = -ai * (a + (offset + i + 1)) * z * z / ((b + (offset + i + 1)) * (z - bi));
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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int offset;
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hypergeometric_1F1_recurrence_2a_and_2b_coefficients operator=(const hypergeometric_1F1_recurrence_2a_and_2b_coefficients&);
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};
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//
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// Recurrence relation for double-stepping on a:
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// -(b-a)(1 + b - a)/(2a-2-b+z)M(a-2,b,z) + [(b-a)(a-1)/(2a-2-b+z) + (2a-b+z) + a(b-a-1)/(2a+2-b+z)]M(a,b,z) -a(a+1)/(2a+2-b+z)M(a+2,b,z)
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//
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template <class T>
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struct hypergeometric_1F1_recurrence_2a_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_2a_coefficients(const T& a, const T& b, const T& z, int offset = 0) :
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a(a), b(b), z(z), offset(offset)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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i *= 2;
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const T ai = a + (offset + i);
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// -(b-a)(1 + b - a)/(2a-2-b+z)
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const T an = -(b - ai) * (b - (a + (offset + i - 1))) / (2 * (a + (offset + i - 1)) - b + z);
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const T bn = (b - ai) * (a + (offset + i - 1)) / (2 * (a + (offset + i - 1)) - b + z) + (2 * ai - b + z) + ai * (b - (a + (offset + i + 1))) / (2 * (a + (offset + i + 1)) - b + z);
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const T cn = -ai * (a + (offset + i + 1)) / (2 * (a + (offset + i + 1)) - b + z);
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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int offset;
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hypergeometric_1F1_recurrence_2a_coefficients operator=(const hypergeometric_1F1_recurrence_2a_coefficients&);
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};
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//
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// Recurrence relation for double-stepping on b:
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// b(b-1)^2(b-2)/((1-b)(2-b-z)) M(a,b-2,z) + [zb(b-1)(b-1-a)/((1-b)(2-b-z)) + b(1-b-z) + z(b-a)(b+1)b/((b+1)(b+z)) ] M(a,b,z) + z^2(b-a)(b+1-a)/((b+1)(b+z)) M(a,b+2,z)
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//
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template <class T>
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struct hypergeometric_1F1_recurrence_2b_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_2b_coefficients(const T& a, const T& b, const T& z, int offset = 0) :
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a(a), b(b), z(z), offset(offset)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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i *= 2;
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const T bi = b + (offset + i);
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const T bi_m1 = b + (offset + i - 1);
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const T bi_p1 = b + (offset + i + 1);
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const T bi_m2 = b + (offset + i - 2);
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const T an = bi * (bi_m1) * (bi_m1) * (bi_m2) / (-bi_m1 * (-bi_m2 - z));
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const T bn = z * bi * bi_m1 * (bi_m1 - a) / (-bi_m1 * (-bi_m2 - z)) + bi * (-bi_m1 - z) + z * (bi - a) * bi_p1 * bi / (bi_p1 * (bi + z));
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const T cn = z * z * (bi - a) * (bi_p1 - a) / (bi_p1 * (bi + z));
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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int offset;
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hypergeometric_1F1_recurrence_2b_coefficients operator=(const hypergeometric_1F1_recurrence_2b_coefficients&);
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};
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//
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// Recurrence relation for a+ b-:
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// -z(b-a)(a-1-b)/(b(a-1+z)) M(a-1,b+1,z) + [(b-a)(a-1)b/(b(a-1+z)) + (2a-b+z) + a(b-a-1)/(a+z)] M(a,b,z) + a(1-b)/(a+z) M(a+1,b-1,z)
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//
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// This is potentially the most useful of these novel recurrences.
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// - - + - +
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template <class T>
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struct hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients
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{
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typedef boost::math::tuple<T, T, T> result_type;
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hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients(const T& a, const T& b, const T& z, int offset = 0) :
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a(a), b(b), z(z), offset(offset)
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{
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}
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result_type operator()(boost::intmax_t i) const
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{
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const T ai = a + (offset + i);
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const T bi = b - (offset + i);
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const T an = -z * (bi - ai) * (ai - 1 - bi) / (bi * (ai - 1 + z));
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const T bn = z * ((-1 / (ai + z) - 1 / (ai + z - 1)) * (bi + z - 1) + 3) + bi - 1;
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const T cn = ai * (1 - bi) / (ai + z);
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return boost::math::make_tuple(an, bn, cn);
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}
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private:
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const T a, b, z;
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int offset;
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hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients operator=(const hypergeometric_1F1_recurrence_a_plus_b_minus_coefficients&);
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};
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#endif
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template <class T, class Policy>
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inline T hypergeometric_1F1_backward_recurrence_for_negative_a(const T& a, const T& b, const T& z, const Policy& pol, const char* function, int& log_scaling)
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{
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BOOST_MATH_STD_USING // modf, frexp, fabs, pow
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boost::intmax_t integer_part = 0;
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T ak = modf(a, &integer_part);
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//
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// We need ak-1 positive to avoid infinite recursion below:
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//
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if (0 != ak)
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{
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ak += 2;
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integer_part -= 2;
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}
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if (-integer_part > static_cast<boost::intmax_t>(policies::get_max_series_iterations<Policy>()))
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return policies::raise_evaluation_error<T>(function, "1F1 arguments sit in a range with a so negative that we have no evaluation method, got a = %1%", std::numeric_limits<T>::quiet_NaN(), pol);
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T first, second;
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if(ak == 0)
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{
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first = 1;
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ak -= 1;
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second = 1 - z / b;
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}
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else
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{
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int scaling1(0), scaling2(0);
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first = detail::hypergeometric_1F1_imp(ak, b, z, pol, scaling1);
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ak -= 1;
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second = detail::hypergeometric_1F1_imp(ak, b, z, pol, scaling2);
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if (scaling1 != scaling2)
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{
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second *= exp(T(scaling2 - scaling1));
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}
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log_scaling += scaling1;
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}
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++integer_part;
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detail::hypergeometric_1F1_recurrence_a_coefficients<T> s(ak, b, z);
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return tools::apply_recurrence_relation_backward(s,
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static_cast<unsigned int>(std::abs(integer_part)),
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first,
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second, &log_scaling);
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}
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template <class T, class Policy>
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T hypergeometric_1F1_backwards_recursion_on_b_for_negative_a(const T& a, const T& b, const T& z, const Policy& pol, const char*, int& log_scaling)
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{
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using std::swap;
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BOOST_MATH_STD_USING // modf, frexp, fabs, pow
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//
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// We compute
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//
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// M[a + a_shift, b + b_shift; z]
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//
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// and recurse backwards on a and b down to
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//
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// M[a, b, z]
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//
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// With a + a_shift > 1 and b + b_shift > z
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//
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// There are 3 distinct regions to ensure stability during the recursions:
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//
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// a > 0 : stable for backwards on a
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// a < 0, b > 0 : stable for backwards on a and b
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// a < 0, b < 0 : stable for backwards on b (as long as |b| is small).
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//
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// We could simplify things by ignoring the middle region, but it's more efficient
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// to recurse on a and b together when we can.
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//
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BOOST_ASSERT(a < -1); // Not tested nor taken for -1 < a < 0
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int b_shift = itrunc(z - b) + 2;
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int a_shift = itrunc(-a);
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if (a + a_shift != 0)
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{
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a_shift += 2;
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}
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//
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// If the shifts are so large that we would throw an evaluation_error, try the series instead,
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// even though this will almost certainly throw as well:
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//
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if (b_shift > static_cast<boost::intmax_t>(boost::math::policies::get_max_series_iterations<Policy>()))
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return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
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if (a_shift > static_cast<boost::intmax_t>(boost::math::policies::get_max_series_iterations<Policy>()))
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return hypergeometric_1F1_checked_series_impl(a, b, z, pol, log_scaling);
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int a_b_shift = b < 0 ? itrunc(b + b_shift) : b_shift; // The max we can shift on a and b together
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int leading_a_shift = (std::min)(3, a_shift); // Just enough to make a negative
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if (a_b_shift > a_shift - 3)
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{
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a_b_shift = a_shift < 3 ? 0 : a_shift - 3;
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}
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else
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{
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// Need to ensure that leading_a_shift is large enough that a will reach it's target
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// after the first 2 phases (-,0) and (-,-) are over:
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leading_a_shift = a_shift - a_b_shift;
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}
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int trailing_b_shift = b_shift - a_b_shift;
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if (a_b_shift < 5)
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{
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// Might as well do things in two steps rather than 3:
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if (a_b_shift > 0)
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{
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leading_a_shift += a_b_shift;
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trailing_b_shift += a_b_shift;
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}
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a_b_shift = 0;
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--leading_a_shift;
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}
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BOOST_ASSERT(leading_a_shift > 1);
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BOOST_ASSERT(a_b_shift + leading_a_shift + (a_b_shift == 0 ? 1 : 0) == a_shift);
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BOOST_ASSERT(a_b_shift + trailing_b_shift == b_shift);
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if ((trailing_b_shift == 0) && (fabs(b) < 0.5) && a_b_shift)
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{
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// Better to have the final recursion on b alone, otherwise we lose precision when b is very small:
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int diff = (std::min)(a_b_shift, 3);
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a_b_shift -= diff;
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leading_a_shift += diff;
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trailing_b_shift += diff;
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}
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T first, second;
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int scale1(0), scale2(0);
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first = boost::math::detail::hypergeometric_1F1_imp(T(a + a_shift), T(b + b_shift), z, pol, scale1);
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//
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// It would be good to compute "second" from first and the ratio - unfortunately we are right on the cusp
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// recursion on a switching from stable backwards to stable forwards behaviour and so this is not possible here.
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//
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second = boost::math::detail::hypergeometric_1F1_imp(T(a + a_shift - 1), T(b + b_shift), z, pol, scale2);
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if (scale1 != scale2)
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second *= exp(T(scale2 - scale1));
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log_scaling += scale1;
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//
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// Now we have [a + a_shift, b + b_shift, z] and [a + a_shift - 1, b + b_shift, z]
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// and want to recurse until [a + a_shift - leading_a_shift, b + b_shift, z] and [a + a_shift - leadng_a_shift - 1, b + b_shift, z]
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// which is leading_a_shift -1 steps.
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//
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second = boost::math::tools::apply_recurrence_relation_backward(
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hypergeometric_1F1_recurrence_a_coefficients<T>(a + a_shift - 1, b + b_shift, z),
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leading_a_shift, first, second, &log_scaling, &first);
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if (a_b_shift)
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{
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//
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// Now we need to switch to an a+b shift so that we have:
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// [a + a_shift - leading_a_shift, b + b_shift, z] and [a + a_shift - leadng_a_shift - 1, b + b_shift - 1, z]
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// A&S 13.4.3 gives us what we need:
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//
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{
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// local a's and b's:
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T la = a + a_shift - leading_a_shift - 1;
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T lb = b + b_shift;
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second = ((1 + la - lb) * second - la * first) / (1 - lb);
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}
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//
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// Now apply a_b_shift - 1 recursions to get down to
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// [a + 1, b + trailing_b_shift + 1, z] and [a, b + trailing_b_shift, z]
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//
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second = boost::math::tools::apply_recurrence_relation_backward(
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hypergeometric_1F1_recurrence_a_and_b_coefficients<T>(a, b + b_shift - a_b_shift, z, a_b_shift - 1),
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a_b_shift - 1, first, second, &log_scaling, &first);
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//
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// Now we need to switch to a b shift, a different application of A&S 13.4.3
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// will get us there, we leave "second" where it is, and move "first" sideways:
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//
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{
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T lb = b + trailing_b_shift + 1;
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first = (second * (lb - 1) - a * first) / -(1 + a - lb);
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}
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}
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else
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{
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//
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// We have M[a+1, b+b_shift, z] and M[a, b+b_shift, z] and need M[a, b+b_shift-1, z] for
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// recursion on b: A&S 13.4.3 gives us what we need.
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//
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T third = -(second * (1 + a - b - b_shift) - first * a) / (b + b_shift - 1);
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swap(first, second);
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swap(second, third);
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--trailing_b_shift;
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}
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//
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// Finish off by applying trailing_b_shift recursions:
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//
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if (trailing_b_shift)
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{
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second = boost::math::tools::apply_recurrence_relation_backward(
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hypergeometric_1F1_recurrence_small_b_coefficients<T>(a, b, z, trailing_b_shift),
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trailing_b_shift, first, second, &log_scaling);
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}
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return second;
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}
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} } } // namespaces
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#endif // BOOST_HYPERGEOMETRIC_1F1_RECURRENCE_HPP_
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